Some Extremal Problems In Inertia And Eccentric Distance Sum

Inertia theory has some extensive applications in matrix theory, control the-ory and graph theory. Chemical indices of a graph play important roles in study-ing the relation between molecular structure and physical and chemical properties of some compounds. It has been extensively studied by a number of researchers. Some good results were obtained. In this dissertation, we continue to investigate some extremal problems in the inertia of Hermitian matrix (including quaternion Hermitian matrix), the inertia of simple undirected graph and the eccentric dis-tance sum of simple undirected graph, respectively. This dissertation includes three parts.In the first part, we investigate the inertia of Hermitian matrices. This part includes two chapters. We study the extremal inertias of Hermitian Schur complement and the definiteness of Hermitian Schur complement. We consider the extremal inertias of least-square bisymmetric solutions to quaternion matrix equation X A= B and give the expression of the extremal solutions. In addition, we give a sufficient and necessary condition for the matrix equation XA=B to have the maximal, the minimal least-square bisymmetric solution.In the second part, we investigate the inertia of graph. This part is divided into four chapters. We consider the inertias of trees. We investigate the minimal positive inertia of unicyclic graphs on n vertices with given girth and characterize the unicyclic graphs with small positive inertias. We study the inertias of bicyclic graphs and characterize the bicyclic graphs with small positive inertias.In the third part, we investigate the eccentricity distance sum of graphs. This part includes five chapters. We present explicit formulae for the values of eccentric distance sum for the Cartesian product and join, applied to some graphs of chemical interest (like nanotubes and nanotori). Various lower and upper bounds for the eccentric distance sum in terms of other graph invariants including the Wiener index, the degree distance, eccentric connectivity index, independence number, connectivity, matching number, chromatic number are established. we characterize the extremal trees with the maximal, the minimal eccentric distance sum and unicyclic graphs with the minimal, the second minimal eccentric distance sum.